Improper integrals as a puzzle for creeping flow ... - AIP Publishing

Physics of Fluids

LETTER

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Improper integrals as a puzzle for creeping flow around an ellipsoid Cite as: Phys. Fluids 31, 021101 (2019); doi: 10.1063/1.5050510 Submitted: 31 July 2018 • Accepted: 8 September 2018 • Published Online: 18 December 2018 Shiyan Wang (王世彦),

Curtis P. Martin,

and Sangtae Kim (金相泰)a)

AFFILIATIONS Davidson School of Chemical Engineering, Purdue University, 480 W. Stadium Ave., West Lafayette, Indiana 47907, USA a)

Electronic mail: [email protected]

ABSTRACT The general ellipsoid immersed in the quadratic ambient velocity field Hijk xj xk with the origin at the ellipsoid center will ´ law contribution to the hydrodynamic drag because of the ambient pressure gradient. Here we exhibit an additional Faxen reexamine the solution of S. Kim and P. Arunachalam [“The general solution for an ellipsoid in low-Reynolds-number flow,” J. Fluid Mech. 178, 535–547 (1987)] for Stokes flow by focusing more carefully on the system of equations and boundary condition on the disturbance velocity field. All of the unknown coefficients (quadrupole moments) in the disturbance velocity were determined by Kim and Arunachalam from a linear system of equations that arise from matching the quadratic terms in the disturbance and ambient fields. Note that when matching the boundary conditions, they interpreted the constant terms in the disturbance velocity field as a complicated but formal expression for the drag on the ellipsoid. Kim and Arunachalam did not explore this further and instead opted for the simpler expression for the drag from a direct application of the ´ law to the quadratic ambient field. Years later, we provide the resolution of the “mystery” of two seemingly difFaxen ferent expressions for the hydrodynamic drag on the ellipsoid by proving their equivalence. One interesting consequence is the emergence of rather formidable integrals that are identically zero. Uniqueness theorems and related properties for Stokes flow provide a guarantee that the two expressions for the drag are equivalent, but we also provide a direct proof via the properties of elliptic integrals and thus provide formal closure to the problem of the ellipsoid in a quadratic ambient field. Published under license by AIP Publishing. https://doi.org/10.1063/1.5050510

Over the years, many friends and colleagues of Professor Bird have enjoyed the annual holiday puzzles and an assortment of mathematical curiosities (like the volume of the “de-cored” sphere) distributed to challenge our minds. In that spirit, we offer the following mathematical gems to return the favor. Consider the following three improper integrals:

∞

0

∞

0

∞

0

(1)

N2 (t)dt , (a2 + t)(c2 + t)(∆(t))3

(2)

N3 (t)dt , (a2 + t)(c2 + t)(∆(t))3

(3)

Published under license by AIP Publishing

N1 = −4a4 b2 c2 − (2a4 b2 + 2a4 c2 − 10a2 b2 c2 )t + (11a2 b2 + 11a2 c2 − b2 c2 )t2 + (12a2 − 2b2 − 2c2 )t3 − 3t4 ,

(4)

N2 = −4a4 b2 c2 − (2a4 c2 − 2a4 b2 + 2a2 b2 c2 )t + (4a4 + 7a2 b2 + a2 c2 − b2 c2 )t2 + (10a2 + 2b2 )t3 + 3t4 ,

N1 (t)dt , (a2 + t)2 (∆(t))3

Phys. Fluids 31, 021101 (2019); doi: 10.1063/1.5050510

where

N3 = 4a2 b2 c2 t + (a2 b2 + 3a2 c2 + b2 c2 )t2 − 2b2 t3 − 3t4 ,

(5) (6)

and the integrand includes a function that appears in the Carlson form of the elliptic integral,1 ∆(t) =

q

(a2 + t)(b2 + t)(c2 + t).

(7)

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The claim (the puzzle) is that for all real numbers a, b, and c, each of these seemingly formidable integrals is somewhat remarkably identically equal to zero. In the following discussion and in keeping with the spirit of this special issue of the journal, we first provide the fluid mechanical origins of these three mathematical identities. In essence, these results must hold as a direct consequence of ´ law for the hydrodynamic drag on an the so-called Faxen ellipsoid in a quadratic ambient field. The direct mathematical proof (using the properties of elliptic integrals) is then provided. The general solution for Stokes flow past an ellipsoid in a polynomial ambient velocity field was solved by Kim and Arunachalam2 as generalized extensions of the classical Stokes flow solutions of Oberbeck,3 Edwardes,4 and Jeffery5 for an ellipsoid in translation (uniform stream), rotation (constant vorticity), and linear fields, respectively. Their primary conclusion is that an ellipsoid in an nth order ambient field creates a disturbance velocity field in the form of an nth order Stokes multipole plus lower order multipole moments of the same (even or odd) orders as the primary field. Explicit results are provided for the quadratic ambient field, vi∞ = Hijk xj xk (index notation and the Einstein summation convention are used for repeated indices), and the x1 , x2 , and x3 (x, y, z) axes are aligned with the principal directions of the ellipsoid with semi-axes a, b, and c, respectively. As required by the general theory, the disturbance field in a quadratic ambient field consists of Stokes quadrupoles (tensor of rank 3 with moment Pijk ) and a Stokes monopole (i.e., a Stokeslet or monopole of rank 1 with moment Pi ). In their paper, the boundary condition is for a stationary ellipsoid, and requiring all quadratic terms in xi xj in the disturbance velocity to match the corresponding terms in the ambient field provided the necessary linear system of equations to determine the unknown multipole moments. Their velocity representation also has constant terms (i.e., polynomials of degree 0), which provide an expression for the hydrodynamic drag (i.e., the monopole Pi ) on the stationary ellipsoid upon substitution of the quadrupole moments as determined from matching of the quadratic terms. Kim and Arunachalam2 chose not to use this (rather complicated) expression for the drag, instead opting for a “shortcut” of using ´ law as suggested by Brenner and Haber.6 We now the Faxen revisit these two alternative routes for determining the hydrodynamic drag and show that the initial expressions for the drag are not trivially equivalent but require the mathematical identities described in the opening remarks! In the following, we examine these steps in more detail to produce the mathematical identities. Due to the symmetry (with respect to the permutation of the symbols x, y, and z and a, b, and c) of the ellipsoid geometry, the 18 × 18 system of the quadratic flow field (ambient flow parameters Hijk and unknown multipole moments Pijk to be determined) can be further divided into four decoupled systems as described by Kim and Arunachalam:2 one 3 × 3 (without an ambient pressure gradient) and three 5 × 5 (ambient pressure gradient in each coordinate direction)

Phys. Fluids 31, 021101 (2019); doi: 10.1063/1.5050510 Published under license by AIP Publishing

LETTER


systems. For the 3 × 3 system, the stationary particle is already force-free. Therefore, only the 5 × 5 systems exhibit the link between the ambient pressure field, the translational velocity, and the hydrodynamic drag. Here, we focus on the case in which the resulting hydrodynamic drag and pressure gradient on the particle are in the x1 direction, while the other two cases (hydrodynamic drag along the x2 and x3 directions) can be realized by index cycling. On the surface of the ellipsoid, the boundary condition requires that the translational motion of the particle is equal to the sum of the ambient and disturbance flow fields, Ui = vi∞ +vid , and the linear relationship between Pijk and Hijk is obtained by matching coefficients in terms of xj xk . Then, the constant term requires2 15 f P1 Go + a2 K1 + U1 = P1kk a2 K1k + Kk 16πµ 64πµ g + (Pkk1 + Pk1k ) a2k K1k − K1 ,

(8)

where ak , or (a, b, c), are the principal semi-axes of the ellipsoid; ∞ dt (9) Go = q 0 (a2 + t)(b2 + t)(c2 + t) is the definition for the Carlson elliptic integral of the first kind without the factor of (1/2), which is denoted as RF by Carlson and Gustafson;1 the elliptic integrals are as in the work of Kim and Arunachalam,2 ∞ dt Ki = , (10) 2 0 (ai + t)∆(t) ∞ dt Kij = , (11) 2 2 0 (ai + t)(aj + t)∆(t) ∞ dt Kijk = . (12) 2 2 2 0 (ai + t)(aj + t)(ak + t)∆(t) Note that Ki [Eq. (10)] has also been defined as the Carlson elliptic integral of the second kind RD without the factor (3/2).1 ´ law of ellipsoids,7 the force Fi Here, we recall the Faxen (or Pi in this letter) on an arbitrary ellipsoid is equal to Fi = µAij ·

! sinh D ∞ vj − µAij Uj , D

(13)

where the operator sinh D/D is interpreted as a power series in D2 , ∞ D2 D4 sinh D X D2(n−1) = =1+ + +..., (14) D 3! (2n − 1)! 5! n=1

with the operator defined by D2 = a2

2 2 ∂2 2 ∂ 2 ∂ + b + c . ∂x2 ∂y2 ∂z2

(15)

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LETTER

µ is the dynamic viscosity of the fluid and Aij is the resistance matrix. For the quadratic ambient field, Eq. (13), based on Ref. 7, becomes F1 sinh D ∞ χo + αo a2 + U1 = v1 = D 16πµabc

a2k H1kk 3

,

(16)

and C5 before the elliptic integral substitutions and placement over a common denominator, C1 = − 3K1 + 18a2 K11 + b2 K12 + c2 K13 − 15a4 K111 − 3a2 b2 K112 − 3a2 c2 K113 ,

1 2 a H111 + b2 H122 + c2 H133 3 15 f = P111 a2 K11 + K1 + P122 a2 K12 + K2 + P133 a2 K13 + K3 64πµ + 2P111 a2 K11 − K1 + (P212 + P221 ) b2 K12 − K1 g (17) + (P313 + P331 ) c2 K13 − K1 ,

(19)

(21)

Now, each of the five coefficients Ci (i = 1, . . . , 5) is a function of Ki , Kij , and Kijk , and all Ci should be identically equal to zero. After substitution of all elliptic integrals as defined in Eqs. (10)–(12), and placing all integrands over a common denominator, we obtain the five improper integrals that must be identically zero. However, two of the integrals (C2 , C4 ) correspond to the cases of interchanging b and c for Eqs. (2) and (3), respectively. Here, the expressions are provided for C1 , C3 ,


2 2

− 3a2 c2 K133 ,

(23)

C5 = − 3K1 + 3a2 K11 + b2 K12 + 6c2 K13 − 3a2 c2 K113 − b2 c2 K123 − 3c4 K133 .

(24)

Note that expressions for C1 , C3 , and C5 are mathematically identical to Eqs. (1), (2), and (3), respectively. Next, we proceed to the mathematical proof of Ci = 0. We employ the following relationships for elliptic integrals: K12 =

K1 − K2 , b2 − a2

K13 =

K23 =

K2 − K3 , c2 − b2

K112 =

K113 =

K11 − K13 , c2 − a2

K133 =

K13 − K33 , c2 − a2

By utilizing Eqs. (18)–(20), Eq. (17) can be written in terms of Pijk ,

Phys. Fluids 31, 021101 (2019); doi: 10.1063/1.5050510

4

K1 − K3 , c2 − a2 K11 − K12 , b2 − a2

K122 =

K12 − K22 , b2 − a2

K123 =

(25)

K13 − K23 . b2 − a2

∞

dt q 0 (t + a2 ) (t + a2 )(t + b2 )(t + c2 ) u1 g (snu)2 du (a2 > b2 > c2 > 0), = 2 (a − c2 ) 0

K1 =

! ! 1 3 1 1 32πµ H133 = − K13 + a2 K113 P111 + K23 + a2 K123 P122 2 2 2 2 15 ! ! 3 3 1 1 + K33 + a2 K133 P133 + − K13 + b2 K123 2 2 2 2 ! 3 3 2 × (P212 + P221 ) + − K13 + c K133 (P313 + P331 ). (20) 2 2

C1 P111 + C2 P122 + C3 P133 + C4 (P212 + P221 ) + C5 (P313 + P331 ) = 0.

2

Therefore, Ci can be further simplified as Ci = Ci (K1 , K2 , K3 , K11 , K22 , K33 , K111 ). These remaining elliptic integrals can be expressed as Jacobi elliptic functions. For instance, K1 can be represented as

!

32πµ 3 3 3 1 K22 + a2 K122 P122 H122 = − K12 + a2 K112 P111 + 2 2 2 2 15 ! ! 1 1 2 3 3 + K23 + a K123 P133 + − K12 + b2 K122 2 2 2 2 ! 1 1 2 × (P212 + P221 ) + − K12 + c K123 (P313 + P331 ), 2 2

2

C3 = 3K3 + 4a K13 − b K23 − 3c K33 − 3a K113 − a b K123

where the ambient flow parameters Hijk and multipole moments Pijk already satisfy (from the boundary condition) ! ! 1 15 3 9 32πµ H111 = − K11 + a2 K111 P111 + − K12 + a2 K112 P122 2 2 2 2 15 ! ! 1 3 2 3 3 + − K13 + a K113 P133 + − K11 + b2 K112 2 2 2 2 ! 3 3 2 × (P212 + P221 ) + − K11 + c K113 (P313 + P331 ), (18) 2 2

!

(22)

2

where the parameter relationships are χ o = abcGo , αo = abcK1 , and the hydrodynamic drag is P1 = F1 . Combining Eqs. (8) and ´ law is (16), the relationship for consistency with the Faxen


(26)

where g = √ 22 2 ; snu is the Jacobi elliptic function (snu a −c = sin φ), where both u and φ satisfy u=

φ

0

dθ p

1 − k2 sin2 θ

,

(27)

and k2 = (a2 − b2 )/(a2 − c2 ); (snu)2 = (a2 − c2 )/(a2 + t). Note thatqk2 < 1. The upper limit u1 can be derived as u1 = sn−1 [ (a2 − c2 )/a2 ]. Similarly, other elliptic integrals are given below,8

K2 =

g (a2 − c2 )

K3 =

g (a2 − c2 )

K11 =

g (a2 − c2 )2

u1

(sdu)2 du,

(28)

(tnu)2 du,

(29)

0 u1 0

u1

(snu)4 du,

(30)

0

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LETTER

K22 = K33 =

g (a2 − c2 )2 (a2

g − c2 )2

u1

B6 M2 M4 J2 J4

where E1 =

u1

u1

(sdu)4 du,

0 u1

(tnu)4 du,

(32)

0

g − c2 )3

u1

(snu)6 du.

a2 − c2 , a2

cnu1 =

q

1 − (snu1 )2 =

c , a

q b cnu1 c 1 − k2 (snu1 )2 = , cdu1 = = , a b dnu1 1 a a 1 ndu1 = = , ncu1 = = , cnu1 c b dnu1 s snu1 a2 − c2 tnu1 = = . cnu1 c2 dnu1 =

Furthermore, integrals of Jacobi elliptic functions can be further expressed as8

We summarize as concluding remarks that the Stokes drag on an ellipsoid in a quadratic ambient flow field, as calcu´ law, is seemingly at odds with the expreslated with the Faxen sion obtained directly from the boundary condition applied to the constant terms in the velocity representation. This difference is resolved only if three improper integrals with free parameters (a, b, c) are identically equal to zero. Of course

Phys. Fluids 31, 021101 (2019); doi: 10.1063/1.5050510

(34)

(35)

The notation for the “basis parameters” (Bi , Mi , Ji ) in Eq. (34) is as defined by Byrd and Friedman.8 The three expressions for C1 , C3 , and C5 only contain basis parameters (Bi , Mi , Ji ), and further reductions using the identities given in Eqs. (34) and (35) reveal exact cancellation of all terms. Therefore, the three improper integrals are indeed identically equal to zero.


(33)

0

we realize that the theory and properties of Stokes flow provide a circuitous proof; nevertheless, a direct proof was established by converting all elliptic integrals to their Jacobi forms. Finally, as our closing remarks, we note that the differentiation and integration of these three identities with respect to the parameters (a, b, c) will generate additional integrals of increasing complexity, all of which are equal to zero.

(dnu)2 du and

snu1 =

(a2

(snu)2 du =

0

s

K111 =

(31)

u1 − E1 , k2 0 u1 1 = (snu)4 du = 4 [(2 + k2 )u1 − 2(1 + k2 )E1 + k2 snu1 cnu1 dnu1 ], 3k 0 u1 (snu1 )3 cnu1 dnu1 + 4(1 + k2 )B4 − 3B2 = (snu)6 du = , 5k2 0 u1 E1 − (1 − k2 )u1 − k2 snu1 cdu1 , = (sdu)2 du = k2 (1 − k2 ) 0 u1 2(2k2 − 1)E1 + (1 − k2 )(2 − 3k2 )u1 − k2 snu1 cdu1 [(1 − k2 )(ndu1 )2 + 4k2 − 2] = (sdu)4 du = , 3k4 (1 − k2 )2 0 u1 dnu1 tnu1 − E1 = (tnu)2 du = , 1 − k2 0 u1 2(2 − k2 )E1 − (1 − k2 )u1 + tnu1 dnu1 [(1 − k2 )(ncu1 )2 − 4 + 2k2 ] , = (tnu)4 du = 3(1 − k2 )2 0

B2 = B4


REFERENCES 1

B. C. Carlson and J. L. Gustafson, “Asymptotic approximations for symmetric elliptic integrals,” SIAM J. Math. Anal. 25, 288–303 (1994). 2 S. Kim and P. Arunachalam, “The general solution for an ellipsoid in lowReynolds-number flow,” J. Fluid Mech. 178, 535–547 (1987). 3 A. Oberbeck, “On steady-state flow under consideration of inner friction,” J. Reine Angew. Math. 81, 62–80 (1876). 4 D. Edwardes, “Steady motion of a viscous liquid in which an ellipsoid is constrained to rotate about a principal axis,” Q. J. Pure Appl. Math. 26, 70–78 (1892). 5 G. B. Jeffery and L. N. G. Filon, “The motion of ellipsoidal particles immersed in a viscous fluid,” Proc. R. Soc. A 102, 161–179 (1922). 6 H. Brenner and S. Haber, “Symbolic operator representation of generalized ´ relations,” PhysicoChem. Hydrodyn. 4, 271–278 (1983). Faxen 7 S. Kim and S. J. Karrila, Microhydrodynamics: Principles and Selected Applications (Butterworth-Heinemann, 1991). 8 P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists (Springer, 2013), Vol. 67.

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